``Particle Systems with Several Conservation Laws: Fluctuations
and Hydrodynamic Limit'' connects different fields where intimate
connections are just emerging. In many applications (like traffic
flow, dust models in astrophysics, compressible fluid models) very
natural microscopic descriptions of the stochastic dynamics of
interacting particles can often be related to macroscopic
continuum descriptions using nonlinear evolutionary PDEs. It is
hard to rigorously relate these two levels of modelling of the
same physical or biological phenomena, though.
Scientific progress in the area of hyperbolic conservation laws
for systems of one and two equations has been used in rigorously
proving the hydrodynamic limit of corresponding interacting
particle systems. Techniques like the theory of compensated
compactness in PDEs are emerging as powerful tools in the
interacting particle system community. Many other original ideas
were and are currently being developed within this second context
for systems with two and more conservation laws. This had lead the
organizers to believe that time has come to devote a high profile
meeting to this subject which is situated at the intersection
between nonlinear hyperbolic pde theory, probability theory of
interacting particle systems, nonequilibrium statistical physics.
More specifically, the choice of the topic was motivated by the
following closely related issues: As it is well-known solutions of
systems of hyperbolic PDEs develop shocks and this fact causes
major difficulties in the mathematical analysis as well as in the
physical interpretation of the microscopic particle structure of a
shock. Moreover, in the presence of macroscopic currents, boundary
conditions in finite systems determine the bulk behaviour of
stationary solutions both of PDEs and particle systems. This has
been shown to lead to boundary-induced nonequilibrium analogs of
phase transitions which are novel phenomena of particular
importance in applications which usually deal with effectively
finite systems. It raises the question how microscopic laws of
interaction find an appropriate description in terms of boundary
conditions of an associated hyperbolic PDE. In our current but not
fully developed understanding, the hydrodynamic limit, existence
of shocks, and the nature of boundary conditions appear to be very
intricately linked problems which require investigation within a
common framework. In this context the workshop was concerned with
the following problems:
\begin{itemize}
\item Derivation of hydrodynamic limit
\item Microscopic structure of the shocks
\item Open boundary problems
\item Dynamical phase transitions
\item Large deviations
\item Treatment of the theory of conservation laws with entropies
coming form microscopic models
\end{itemize}
The participants, coming from the US, France, Hungary and Germany,
were mathematicians from PDE theory and probability theory and
physicists working in the field of nonequilibrium statistical
mechanics. With all of them being specialists coming from
different fields, but sharing a common research interest, this
miniworkshop turned out to be a highly fruitful ``joint venture''.
A number of very successful expository lectures on recent progress
in the field helped to bridge the gaps between the different
communities. More specialized talks, partly on open problems, led
the participants to leave the confines of their respective
communities and to interact with each other. All of us enjoyed
enormously the externally tranquil, but scientifically vivid and
stimulating atmosphere of Oberwolfach.